The symmetric-product semi-orthogonal decomposition conjecture

Let $C$ be a projective smooth curve of genus $g\geq 2$, and let $C_d$ denote its $d$-th symmetric product. Let $D(C_d)$ be its derived category, and call a semi-orthogonal decomposition non-trivial if it is not the trivial one. Symmetric-product conjecture. There is no non-trivial semi-orthogonal decomposition on $D(C_d)$ for

$$1\leq d\leq g-1.$$

The source proves this in several ranges, including $d<\operatorname{gon}(C)$ and $d=2$ for curves of genus at least $3$, while the full range remains open in the supplied source. The statement was independently stated by Belmans, Galkin and Mukhopadhyay.