Your conjecture is true. We assume PD throughout. The proof I see requires citing a number of facts from inner model theory and descriptive set theory. First, it uses Woodin's theorem characterizing the reals of $M_{2n-1}$ as the set $Q_{2n+1}$ of points in Baire space that are $\Delta^1_{2n+1}$ definable from a countable ordinal. In symbols:
Theorem (Woodin). $\omega^\omega\cap M_{2n-1} = Q_{2n+1}$.
It uses a correctness theorem for the odd levels:
Theorem $M_{2n-1}$ is $\Pi^1_{2n}$-correct.
A set $A\subseteq \omega^\omega$ is $\Pi^1_{2n+1}$-bounded if $\Pi^1_{2n+1} = \exists^{A} \Pi^1_{2n+1}$. We need that $Q_{2n+1}$ is $\Pi^1_{2n+1}$-bounded. In fact, something stronger is true (see Kechris-Martin-Solovay's "Introduction to $Q$-theory"):
Theorem (Kechris-Martin-Solovay). $Q_{2n+1}$ is the largest $\Pi^1_{2n+1}$-bounded subset of Baire space.
We need Moschovakis's "Spector-Gandy theorem for the odd levels" (Moschovakis, Descriptive Set Theory, 6E.7):
Theorem (Moschovakis) $\Pi^1_{2n+1}\cap\omega^\omega = \exists^{\Delta^1_{2n+1}\cap \omega^\omega}\Pi^1_{2n}\cap \omega^\omega$.
Moschovakis's theorem will actually be applied to $Q_{2n+1}$ using the $Q$-Theory Reflection Theorem:
Theorem (Kechris-Martin-Solovay) If $A\subseteq \omega^\omega$ is $\Pi^1_{2n+1}$, then $\exists x\in \Delta^1_{2n+1}\ A(x)$ if and only if $\exists x\in Q_{2n+1}\ A(x)$.
Given these facts, the calculation becomes a straightforward pointclass calculation.
By definition, $\Sigma^1_{2n+1} = \exists^{\omega^\omega}\Pi^1_{2n}$, so $(\Sigma^1_{2n+1})^{M_{2n-1}} = \exists^{\omega^\omega\cap M_{2n-1}}(\Pi^1_{2n})^{M_{2n-1}}$.
Woodin's theorem characterizing $\Pi^1_{2n+1}$ along with the $\Pi^1_{2n}$-correctness of $M_{2n-1}$ imply $\exists^{\omega^\omega\cap M_{2n-1}}(\Pi^1_{2n})^{M_{2n-1}}\cap\omega^\omega = \exists^{Q_{2n+1}}\Pi^1_{2n} \cap\omega^\omega$.
Moschovakis's Spector-Gandy Theorem along with the $Q$-Theory Reflection Theorem yields that $\exists^{Q_{2n+1}}\Pi^1_{2n}\cap \omega^\omega = \exists^{\Delta^1_{2n+1}\cap\omega^\omega}\Pi^1_{2n}\cap \omega^\omega = \Pi^1_{2n+1}\cap \omega^\omega$.
Stringing together a bunch of pointclass identities, one can conclude that $(\Sigma^1_{2n+1})^{M_{2n-1}}\mathrel{\cap} \omega^\omega = \Pi^1_{2n+1} \cap \omega^\omega$.
You might also want to look at Theorem 4.12 of John Steel's paper Projectively Well-Ordered Inner Models. I think you can use the proof to get a more inner model theoretic proof of your conjecture, but Steel's result is closely related and of independent interest.