Shannon capacity $\theta(G)$ of pentagon is achieved at $2$-fold strong product of the pentagon.

It is also known that the Lov'asz theta $\vartheta(G)^m\neq\alpha(G^{\boxtimes m})$ for any finite positive integer $m$ if $G$ is an odd cycle of length $>5$.

Is it known that $\theta(G)^m\neq\alpha(G^{\boxtimes m})$ for any finite positive integer $m$ if $G$ is an odd cycle of length $>5$? That is $\limsup_{m\rightarrow\infty}\alpha(G^{\boxtimes m})^{\frac{1}{m}}$ is not attained at a finite positive integer $m$.

Note that the the above statement is true would still not be strong enough to decide $\vartheta(G)=\theta(G)$.

Shannon capacity $\Theta(G)$ of pentagon is achieved at $2$-fold strong product of the pentagon.

It is also known that the Lov'asz theta $\vartheta(G)^m\neq\alpha(G^{\boxtimes m})$ for any finite positive integer $m$ if $G$ is an odd cycle of length $>5$.

Is it known that $\Theta(G)^m\neq\alpha(G^{\boxtimes m})$ for any finite positive integer $m$ if $G$ is an odd cycle of length $>5$? That is $\limsup_{m\rightarrow\infty}\alpha(G^{\boxtimes m})^{\frac{1}{m}}$ is not attained at a finite positive integer $m$.

Note that the the above statement is true would still not be strong enough to decide $\vartheta(G)=\Theta(G)$.