Even-index period formula conjecture for the family $1(01)^k$

Consider the family of finite configurations whose current binary word is $1(01)^k$, with period function $P(k)$ assigning to the configuration indexed by $k$ its period.

Even-index period formula conjecture. The period function satisfies

$$P(2i)=\frac{2\cdot4^{2i+2}-7\cdot4^{i+1}+5}{3}$$

for $i\in\mathbb{N}$. This formula is inferred from experimentally observed base-$4$ patterns for the periods at even indices; the source presents it as a conjectural formula and gives no proof or resolution.