Maximal-rank restriction conjecture for linear systems on blow-ups

Let $X$ be the blow-up of $\mathbb P^2$ at $r$ general points, let $D$ be an effective divisor on $X$, and let $E$ be a $(-1)$-curve. Consider the restriction map

$$\lvert D\rvert\xrightarrow{\rho}\lvert D_{\vert E}\rvert.$$

The conjecture asserts: (1) all base points in the image of $\rho$ come from a base curve in $\lvert D\rvert$; and (2) if $\lvert D\rvert$ has no fixed component, then $\rho$ has maximal rank. This predicts maximal-rank behavior for restrictions of plane linear systems to exceptional curves; the source gives no resolution status.