Positive Divisors and Möbius Function

Let $\mu(n)$ denote the Möbius function, and let $d(n)$ denote the number of positive divisors of $n$. For every positive integer $n$, define the arithmetic function $\mathcal{F}: \mathbb{N}^+ \to \mathbb{N}^+$ by $$\mathcal{F}(n) = \prod_{d \mid n} \left( \operatorname{lcm}\left(d, \left\lfloor \frac{n}{d} \right\rfloor \right) \right)^{\mu(n/d)}$$and let $\mathcal{G}: \mathbb{N}^+ \to \mathbb{N}^+$ be given by$$\mathcal{G}(n) = \sum_{d \mid n} \mathcal{F}(d).$$It is known that $\mathcal{G}(p)$ is the same for every prime $p$. Find the value of $\mathcal{G}(p)$.