Strong existence for SDEs with singular drifts driven by fractional Brownian motion

Let $B^H$ be a $d$-dimensional fractional Brownian motion with Hurst parameter $H\in(0,1/2)$. We prove the existence of a strong solution to $$d X_t=b(t,X_t)d t+d B_t^H, X_0=x_0\in R^d,$$when $b\in L^q([0,T];L^p(\mathbb R^d;\mathbb R^d)),2\le p,q<\infty$ and the subcritical Ladyzhenskaya--Prodi--Ser