Let $C$ be a curve and $L$ be a line bundle with sufficiently large degree. Let $C_p$ denote the $p$-th symmetric product of $C$, which consists of all the effective divisors of degree $p$ on $C$. Let $pr:C\times C_p\to C$ be the projection and $\sigma:C\times C_p\to C_{p+1}$ defined by $(x,\xi_p)\mapsto x+\xi_p$. Let $E_L$ be the locally free sheaf $\sigma_*pr^*\mathcal{O}(L)$ on $C_{p+1}$. Let $B$ be the projective bundle $\mathbb{P}(E_L)$ with the natural projection $\pi:B\to C_{p+1}$. Is it true that $H^i(B,\mathcal{O}(d))=0$ for all $0<i<2p+1$ and $d$ an arbitrary integer?

By some results by Ein and others(see Singularities and syzygies of secant varieties of nonsingular projective curves), $\mathcal{O}(1)$ maps $B$ to the $p$-th secant variety $\Sigma_p$ of $C$ in $\mathbb{P}H^0(C,L)$ and we have $H^i(\Sigma_p,\mathcal{O}(d))=0$ for all $1<i<2p+1$ and integers $d$. I think if we can further generalize the vanishing result to $B$, it helps to understand the syzygies of secant varieties of curves.