Failure of the pointwise-convergence hypothesis in the non-compact sufficiency theorem

Let $(X,d)$ be a non-compact metric space, let $f:X\to\mathbb{R}$ be bounded and continuous, and let $p:X\times\mathbb{R}\to\mathbb{R}$ be a convex extension of $f$. Suppose there exist maximizing sequences $\{x_n\}$ and $\{y_n\}$ of $f$, a $\delta>0$, and sequences $t_n\to0+$ and $s_n\to0-$ such that

$$\limsup_{n\to\infty}p_{t+}(x_n,t_n)\geq0,\qquad \liminf_{n\to\infty}p_{t-}(y_n,s_n-)\leq0.$$

Pointwise-convergence failure conjecture. The conclusion that $p$ is a Birkhoff-James extension fails if the assumption that $p(x_n,\,\rule{3mm}{0.15mm})$ and $p(y_n,\,\rule{3mm}{0.15mm})$ converge pointwise to a common function $g$ on $(-\delta,\delta)$ is removed.

The claim asserts that the pointwise-convergence hypothesis in the preceding sufficient condition is essential: without it, the remaining assumptions need not force $p$ to be a Birkhoff-James extension. The source provides no resolution, so the conjecture is recorded as open.