Concavity conjecture for the generalized inverse hyperbolic sine

For $x\in(0,1)$ fixed, let $p\mapsto {\rm arcsinh}_p(x)$ denote the generalized inverse hyperbolic sine as a function of $p$. Concavity conjecture. The function $p\mapsto {\rm arcsinh}_p(x)$ is strictly concave on $(0,\infty)$. In particular, for all $p>1$ and $x\in(0,1)$,

$${\rm arcsinh}_p^2(x)>{\rm arcsinh}_{p-1}(x){\rm arcsinh}_{p+1}(x).$$

This conjecture is motivated by computer experiments; the asserted strict concavity would imply the displayed Turán-type inequality.