Effective equidistribution for some unipotent flows in $PSL(2,\mathbb{R}{)}^k$ mod cocompact, irreducible lattice

Let $d\ge 2$, and let $\mathrm{\Gamma }\subset PSL(2,\mathbb{R}{)}^d$ be an irreducible, cocompact lattice. We prove a sharp estimate up to a logarithmic factor on the rate of equidistribution of coordinate horocycle flows on $\Gamma \backslash PSL(2, $\mathbb{R}$)^d$.