  We study the structure of the finite-dimensional representations of
$\mathfrak{sl}_2[t]$, the current Lie algebra type of $A_1$, which are obtained
by taking tensor products of special Demazure modules. We show that these
representations admit a Demazure flag and obtain a closed formula for the
graded multiplicities of the level 2 Demazure modules in the filtration of the
tensor product of two local Weyl modules for $\mathfrak{sl}_2[t]$. Furthermore,
we derive an explicit expression for graded character of the tensor product of
a local Weyl module with an irreducible $\mathfrak{sl}_2[t]$ module. In
conjunction with the results of \cite{MR3210603}, our findings provide evidence
for the conjecture in \cite{9} that the tensor product of Demazure modules of
levels m and n respectively has a filtration by Demazure modules of level m +
n.
