Based on a later edit and comment on this question:

Considering only the accented melody notes in Bars 13-16, it's actually easier to argue the melody is in F sharp minor than G sharp anything:

• E sharps are plentiful and are always immediately followed by F sharps, strongly implying that the E sharps are leading tones and therefore F sharp is the tonic
• The melody can be plausibly harmonized as C♯-F♯m-C♯-F♯m / C♯m-G♯-C♯m-C♯m / C♯-F♯m-C♯-F♯m / C♯- F♯m-A-C♯ (this obviously isn't what Chopin actually used, though)
• Taken alone, the melody notes used are F♯-G♯-A-C♯-D♯-E-E♯; this collection cannot be G sharp Phrygian because it contains E sharp

Zoom out and consider all melody notes in Bars 13-16 and a G sharp tonic becomes even more implausible:

• The melody notes used are G♯-A-B-B♯-C♯-D♯-E-E♯-F♯, already implying great chromaticism and a lack of modal flavour due to the presence of 3 notes in a row separated by a semitone each
• Especially given that minor-key music often uses a sharpened leading tone instead of a flattened subtonic for the 7th scale degree, taking the voice leading of the full melody into consideration, tonics of F sharp, C sharp, and E are more plausible than G sharp, which is only ever preceded by E, A (accented melody notes edition), E, and B (full melody edition) - never F double sharp or even F sharp like we likely expect for a G sharp tonic
• With 9 G sharps and 12 F sharps in the full melody, F sharp is more common than G sharp in the full melody, further harming the case for G sharp as the tonic
• The full melody can be plausibly harmonized as C♯-F♯m-C♯-F♯m / F♯m-D♯°-E-E / C♯-F♯m-C♯-F♯m / E♯ half-diminished 7th-B-Esus4-E, actually making the case that E is the tonic instead as the music rests on E major chords the most often and all E major and E major-like chords are immediately preceded by dominant-function chords in E major or just more E major(-like) chords

Overall, the requested melody of Chopin's Fantaisie-Impromptu contains too many notes not in G sharp Phrygian to be in G sharp Phrygian, the melody notes do not undergo voice leading consistent with G sharp Phrygian, and analyzing the harmony of the requested melody alone reveals that it is in E major instead with heavy tonicization of F sharp minor and C sharp.

Based on a later edit and comment on this question:

Considering only the accented melody notes in Bars 13-16, it's actually easier to argue the melody is in F sharp minor than G sharp anything:

• E sharps are plentiful and are always immediately followed by F sharps, strongly implying that the E sharps are leading tones and therefore F sharp is the tonic
• The melody can be plausibly harmonized as C♯-F♯m-C♯-F♯m / C♯m-G♯-C♯m-C♯m / C♯-F♯m-C♯-F♯m / C♯- F♯m-A-C♯ (this obviously isn't what Chopin actually used, though)
• Taken alone, the melody notes used are F♯-G♯-A-C♯-D♯-E-E♯; this collection cannot be G sharp Phrygian because it contains E sharp

Zoom out and consider all melody notes in Bars 13-16 and a G sharp tonic becomes even more implausible:

• The melody notes used are G♯-A-B-B♯-C♯-D♯-E-E♯-F♯, already implying great chromaticism and a lack of modal flavour due to the presence of 3 notes in a row separated by a semitone each
• Especially given that minor-key music often uses a sharpened leading tone instead of a flattened subtonic for the 7th scale degree, taking the voice leading of the full melody into consideration, tonics of F sharp, C sharp, and E are more plausible than G sharp, which is only ever preceded by E, A (accented melody notes edition), E, and B (full melody edition) - never F double sharp or even F sharp like we likely expect for a G sharp tonic
• With 9 G sharps and 12 F sharps in the full melody, F sharp is more common than G sharp in the full melody, further harming the case for G sharp as the tonic
• The full melody can be plausibly harmonized as C♯-F♯m-C♯-F♯m / F♯m-D♯°-E-E / C♯-F♯m-C♯-F♯m / E♯ half-diminished 7th-B-Esus4-E, actually making the case that E is the tonic instead as the music rests on E major chords the longest and all E major and E major-like chords are immediately preceded by dominant-function chords in E major or just more E major(-like) chords

Overall, the requested melody of Chopin's Fantaisie-Impromptu contains too many notes not in G sharp Phrygian to be in G sharp Phrygian, the melody notes do not undergo voice leading consistent with G sharp Phrygian, and analyzing the harmony of the requested melody alone reveals that it is in E major instead with heavy tonicization of F sharp minor and C sharp.