Replaced earlier answer, which used an incorrect definition of interval filament graphs.

Unipolar graphs are interval filaments graphs. Brief sketch: If $G$ is unipolar there are vertex sets $A$ and $B$ s.th. $A$ induces a clique and $B$ induces a set of cliques with vertex sets $B_1, \ldots, B_k$, say.

Each filament has two endpoints on the x-axis, and we order the left endpoints to be consistent with the grouping $A, B_1, \ldots, B_k$, and then place the right endpoints in the reverse order to the right of $B_k$. 

Because the right endpoints are in the reverse order of the left endpoints, we can draw all filaments simultaneously as $/\backslash$-shaped curves and so that no two of the filaments intersect. To create an intersection between an $A$-filament $a$ and a $B_i$-filament $b$, let $x$ be a coordinate inside the group $B_i$, where both $a$ and $b$ are present. We can then push $a$ up (close to $x$) until it is above all the other $B_i$-filaments at $x$, and push $a$ down, until it it below all the other $A$-filaments at $x$, and then make $a$ and $b$ intersect. Creating intersections between two filaments belonging to the same group is similar (and easier).