Linear Toffoli-gate bound for simple periodic-function synthesis

Let $p$ be an odd period represented by an $n$-bit binary number, and let $S_p$ be a simple periodic function, namely a periodic injective function whose synthesized circuit minimizes the number $N_T$ of Toffoli gates.

Toffoli-gate bound conjecture. To synthesize the circuit for $S_p$, one needs at most $n$ Toffoli gates:

$$N_T\leq n.$$

The bound is inferred from circuits computed for odd periods through five bits. Its validity for arbitrary bit-lengths is left conjectural.