Let $f,g:(D^2,j_\mathrm{std})\to(B^{2n}(1),J)$ be two pseudo-holomorphic maps. The following unique continuation result is well-known (it may be proved using either Aronszajn's Lemma or the Carleman similarity principle as in Floer--Hofer--Salamon):

Lemma: Suppose $f=g$ over a neighborhood of $0\in D^2$. Then $f=g$ everywhere.

Is there a similar unique continuation result for pseudo-holomorphic submanifolds (instead of maps)? Here is a precise formulation of what I expect might be true:

Conjecture: Suppose $f=g\circ\phi$ over $D^2(r)$ ($r<1$), for some homeomorphism $\phi:D^2(r)\to D^2(r)$ which is a biholomorphism over $D^2(r)^\circ$ (the interior). Then $\phi$ extends holomorphically to $D^2(r+\epsilon)$ for some $\epsilon>0$ (and hence $f=g\circ\phi$ over $D^2(r+\epsilon)$ by the unique continuation result for maps).

Is some statement along these lines known?

Let $f,g:(D^2,j_\mathrm{std})\to(B^{2n}(1),J)$ be two pseudo-holomorphic maps. The following unique continuation result is well-known (it may be proved using either Aronszajn's Lemma or the Carleman similarity principle as in Floer--Hofer--Salamon):

Lemma: Suppose $f=g$ over a neighborhood of $0\in D^2$. Then $f=g$ everywhere.

Is there a similar unique continuation result for pseudo-holomorphic submanifolds (instead of maps)? Here is a precise formulation of what I expect might be true:

Conjecture: Let $U,V\subseteq(D^2)^\circ$ be two closed topological disks with smooth boundary, and let $\phi:U^\circ\to V^\circ$ be a biholomorphism (which necessarily extends continuously to a homeomorphism $U\to V$). Suppose $f=g\circ\phi$ over $U$. Then $\phi$ extends holomorphically to an open neighborhood of $U$ (and hence $f=g\circ\phi$ over this neighborhood by the unique continuation result for maps).

Is some statement along these lines known?