Equality of convex functions from equal minimal subgradient norms

Let $\mathcal{H}$ be the underlying Hilbert space, and let $\psi_1,\psi_2:\mathcal{H}\rightarrow\mathbb{R}$ be finite convex functions bounded from below. For each $x\in\mathcal{H}$, let $\partial\psi_i(x)$ denote the subdifferential of $\psi_i$ at $x$.

Equal minimal subgradient norm conjecture. If

$$\inf_{p\in\partial\psi_1(x)}\|p\|=\inf_{q\in\partial\psi_2(x)}\|q\|\quad\text{for all }x\in\mathcal{H},$$

then $\psi_1=\psi_2+c$ for some constant $c>0$.

This conjecture would extend the preceding uniqueness theorem for twice continuously differentiable convex functions to finite, possibly nonsmooth convex functions. The source presents it as plausible but does not provide a resolution.