Multiplicativity conjecture for the degree of irrationality of products of general curves

Let $C_1$ and $C_2$ be curves of genera at least $2$ that are very general in moduli. Write $\operatorname{gon}(C_i)$ for the gonality of $C_i$ and $\operatorname{irr}(C_1\times C_2)$ for the degree of irrationality of their product.

Product-of-curves conjecture.

$$\operatorname{irr}(C_1\times C_2)=\operatorname{gon}(C_1)\cdot\operatorname{gon}(C_2).$$

The upper bound is known in general, and equality has been proved for some special curves; the conjecture asserts equality for very general curves of genera at least $2$.