Consider a surface of revolution $S$ with constant positive curvature and an embedding $e:S \hookrightarrow X^3$ for $X^3=[0,1]^3$ with conjugate points $p,q$ anchored on $\partial X^3$ where $\partial X^3=X^3-(0,1)^3$ for $\mathrm{dist}(p,q)=\sqrt{3}$.
What is $\rho_{\mathrm{max}}=\mathrm{max} \lbrace \mathrm{vol}(S) \rbrace_{p,q}$ assuming $S$ must remain a surface of revolution with constant positive curvature?
In other words, what is the volume of the largest surface of revolution with a constant positive curvature metric that can be embedded in $X^3$ anchored at $p,q$?
If you vary the conjugate points $p,q$ you can get an inscribed sphere, whether that volume is maximal amongst all possible $p,q$ I don't know.