Expectation asymptotic for the analysis-to-synthesis operator-norm ratio

Let $A$ be a random matrix in $\mathbb R^{m\times k}$ with independently and identically normally distributed entries, and let $L$ be the synthesis operator appearing in the formulation above. Under these conditions, consider the ratio $\|AL\|_2^2/\|A\|_2^2$. Expectation for the ratio $\frac{\|AL\|_2^2}{\|A\|_2^2}.$ The expectation satisfies

$$\mathbb E\left[\frac{\|AL\|_2^2}{\|A\|_2^2}\right] = \frac{(2k+1)^2}{16\pi^2} + o(1).$$

This is proposed as a tighter asymptotic bound than the preceding lower bound, and numerical experiments suggest that the bound is tight. The source does not establish the conjecture analytically, so its status remains open.