Convexity conjecture for Rényi entropy under symmetric $\kappa$-concavity

Let $(X,Y)$ be a symmetric $\kappa$-concave random vector in $\mathbb{R}^2$, and suppose that $X$ and $Y$ have equal $p$-Rényi entropy. Convexity conjecture for Rényi entropy. The function

$$\lambda\mapsto h_p(\lambda X+(1-\lambda)Y)$$

is convex on $[0,1]$. This statement is presented as stronger than the norm conjecture above. Its proposed convexity would imply the relevant linearized reverse Rényi entropy power inequality, but the source does not report a general proof.