A prime plus a square: A conditional 11/20 upper bound

Let $E_\square(N)$ count the positive integers at most $N$ that cannot be expressed as a prime plus a nonnegative integer square. Assuming that every nontrivial zero of every primitive Dirichlet $L$-function has real part at most $7/8$, we prove $E_\square(N)\ll_\eps N^{11/20+\eps}$ for every $\eps>