Nonexistence conjecture for injective linear series on very general curves

Let $X$ be a smooth curve of genus $g$, and let an injective linear series $g^2_d$ on $X$ mean a base-point-free, not necessarily complete, two-dimensional linear series of degree $d$ inducing a separable and injective morphism from $X$ to ${\mathbb P}^2$. Nonexistence conjecture. For large $g$, a very general smooth curve of genus $g$ has no injective linear series $g^2_d$. This conjecture concerns the existence of injective plane models of smooth curves and complements the positive constructions obtained from cuspidal projections and zero-dimensional schemes; the source does not specify a resolution.