The definition of pseudo-Anosov homeomorphisms on a surface-with-boundary is not as well established as it is on a closed surface. Your question exhibits one reason for this: however you might want to formulate it, this semiconjugacy statement is false on a surface with boundary.

Here's why. Let $S$ be the surface, and list its boundary components as $\partial S = b_1 \cup\cdots\cup b_K$. Let $S/\partial$ be the quotient surface obtained from $S$ by collapsing each $b_k$ to a point denoted $p_k$, and let $P = \{p_1,...,p_K\}$. Whatever it might mean for $g : S \to S$ to be pseudo-Anosov, let us agree that one consequence is that the induced homeomorphism-of-pairs $g / \partial : (S/\partial,P) \to (S/\partial,P)$ satisfies the definition of a pseudo-Anosov homeomorphism on a closed surface, *except* that a point of $P$ is allowed to have a one-pronged singularity in its unstable measured foliation (and, thus also, in its stable measured foliation). In the setting of the mapping class group of $S/\partial$ relative to $P$, Handel's theorem is true: if $g / \partial$ is pseudo-Anosov, and if $\vartheta : (S/\partial,P) \to (S/\partial,P)$ is isotopic to $g / \partial$ relative to $P$, and if $g$ and $\vartheta$ have the same topological entropy, then there is a semiconjugacy from $\vartheta$ to $g$. 

But what then happens back in $S$ itself? The problem is that the restriction of $g$ to $\partial S$ does not contribute to the entropy, and anything goes there. To simplify matters, let me assume that $\partial S = b_1$ is connected. In the definition of pseudo-Anosov for a surface-with-boundary that is presented in [FLP] Exposé Eleven, the restriction $g \mid b_1$ has rational rotation number in $\mathbb{R} / \mathbb{Z}$, because $g \mid b_1$ has a finite invariant set, namely the set of singularities. However, one can isotope $g$ to $g'$, without raising the entropy, so that $g' \mid b_1$ has any rotation number whatsoever. But if two circle homeomorphisms have different rotation numbers, then there is no semiconjugacy in either direction.

Perhaps this can be repaired by imposing artificial conditions on $g \mid \partial S$ which enforce a particular topological behavior on (and near) $\partial S$...