Legendre-polynomial condition for increasing equilibrium density

Let $f_d(t)$ denote the expected density of internal equilibrium points at $t>0$ in a $d$-player, two-strategy random evolutionary game. The density is expressed in terms of Legendre polynomials, and Proposition 3.5 gives a sufficient condition, namely

$$(2d+1)P_d^4 \geq P_{d-1}^2\left[(2d-1)P_{d+1}^2+2P_d^2\right],$$

for $f_{d+1}(t)$ to be increasing as a function of $d$.

Increasing-density conjecture. For any given $t>0$, $f_d(t)$ is an increasing function of $d$.

The sufficient inequality is numerically verified in the source, while a rigorous proof is stated to be unclear. The conjecture is motivated additionally by the established asymptotic behavior $f_d(t)\/\sim\sqrt{d-1}$ for sufficiently large $d$.