Conjecture on sup-norms of diagonal pullbacks of Saito–Kurokawa lifts

Let $F\in\mathrm{SK}_k$ be a Hecke eigenform, let $F^\circ$ denote its pullback to the diagonal, and suppose that $F^\circ\ne0$ and $\lVert F^\circ\rVert_2=1$. The Saito–Kurokawa pullback sup-norm conjecture. One should have

$$\lVert F^\circ\rVert_\infty\ll k^{1/2+o(1)}.$$

The conjecture concerns the expected individual sup-norm of a nonzero normalised diagonal pullback of a Hecke-eigenform Saito–Kurokawa lift; the supplied text gives no resolution status.