Existence of strictly polystable extensions for generic line bundles

Let $X$ be a compact Riemann surface of genus $g_X\geq 2$, and let $L$ be a generic line bundle with $\deg L\leq -\frac{1}{2}g_X$. An extension of $L^{-1}$ by $L$ is a short exact sequence

$$0\longrightarrow L\longrightarrow V\longrightarrow L^{-1}\longrightarrow 0.$$

Existence conjecture. There exists a strictly polystable extension of $L^{-1}$ by $L$. This would extend the existence results for reducible cone spherical metrics to generic line bundles in the indicated degree range; the source presents it as an open question for Riemann surfaces of genus at least two.