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 endpoints along the $x$-axis as $A, B_1, B^R_1, B_2, B^R_2, \ldots, B_k, B^R_k, A^R$, where we think of $A$, and $B_i$ as ordered sets, and let $X^R$ denote the reverse of the ordered set $X$. 

We can then 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).