Stability of approximable Kähler geodesic rays under interpolation

Let $\mathcal R$ be the space of bounded weak geodesic rays emanating from $0$. Let $\{u_t^0\},\{u_t^1\}\in\mathcal R$ be approximable rays, and let $s\mapsto\{u_t^s\}$ be the $d_1^c$-geodesic connecting them.

Interpolation conjecture. For each $s\in[0,1]$, the ray $\{u_t^s\}$ is also approximable.

Approximable rays are important in K-stability and are characterized by an $\mathcal I$-model regularity property. Whether this class is preserved by $d_1^c$-geodesic interpolation remains open.