Let $T_1$ and $T_2$ be two Grothendieck topologies on the same small category $C$, and let $T_3 = T_1 ∪ T_2$ (by which I mean the smallest Grothendieck topology on C containing $T_1$ and $T_2$).

Assume that every $T_3$ cover has a refinement of the form $\stackrel{f_1}{\to} \stackrel{f_2}{\to}$ where $f_2$ is a $T_2$ cover and $f_1$ is a $T_1$ cover (this is true for example, if $T_1$ is the Nisnevich topology, and $T_2$ is the topology consisting of proper cdh covers; in this case $T_3$ is the cdh topology).

If $F$ is a $T_1$-sheaf, is its $T_2$-sheafification still a $T_1$-sheaf (and therefore a $T_3$-sheaf)?

I can show this is true if $F$ is a $T_2$ separated $T_1$ sheaf, but I want to remove this condition.