Projective dimension of powers of cover ideal of Ferrers graphs

Let $\lambda =({\lambda }_1,\dots ,{\lambda }_n)$ be a partition with ${\lambda }_1=m$. Denote by $J_{\lambda }$ the cover ideal in the polynomial ring $S=k[x_1,\dots ,x_n,y_1,\dots ,y_m]$ associated to the Ferrers graph corresponding to $\lambda$. Let $d(\lambda )$ denote the number of distinct parts of $\lambda$. We prove that $$\mathrm{pd}(S\mathrm{/}{J}_{\lambda }^t)=\min \{t,d(\lambda )\}+1$$ for all $t\ge 1$.