Maximum-perimeter convex subset conjecture for triangles

Let $\Omega$ be a triangle with vertices $A$, $B$, and $C$. Assume that its diameter is the side $[AB]$ and that $\widehat{CAB}\geq\widehat{CBA}$. For $c\in[0,|\Omega|]$, consider convex subsets $\omega\subset\Omega$ with area $|\omega|=c$. Triangle maximizer conjecture. The solution of

$$\max\{P(\omega)\mid \omega\text{ is convex},\ \omega\subset\Omega,\ |\omega|=c\}$$

is the triangle $MAB$, where $M\in[AC]$ and its area is equal to $c|\Omega|$. This conjecture specifies the optimizer for the perimeter-maximization problem in a triangular container; the supplied evidence marks it as resolved, although the excerpt does not state the result resolving it.