Can an efficiently computable non-one way permutation be written as the composition of polynomially many easy to compute involutions?

Suppose that $p$ is a polynomial. Then does there exist a polynomial $q$ where if $f:\{0,1\}^{n}\rightarrow\{0,1\}^{n}$ is a bijection where both $f$ and $f^{-1}$ are computable by circuits with at most $p(n)$ gates, then there exists involutions $\iota_{1},\dots,\iota_{k}:\{0,1\}^{n}\rightarrow\{0,1\}^{n}$ where $$f=\iota_{k}\circ\dots\circ\iota_{1}$$ and where each $\iota_{i}$ is computable by a circuit with at most $q(n)$ gates and where $k\leq q(n)$.