Asssuming a probability space $(\Omega,(\mathcal{F}_t)_{t\geq 0},\mathbb{P})$ such that $(\mathcal{F}_t)_{t\geq 0}$ is generated by a Brownian motion $W_t$. We assume that $s>0$ is fixed and $t\in[s,s+\Delta]$, where $\Delta>0$ is finite. I am trying to figure out the conditional distribution of the following process $$\int_{s}^t \frac{dW_u}{(t-u)^{\gamma}}\bigg|\mathcal{F}_s$$
where $0<\gamma<1/2 $. In this case, Itô's lemma does not apply because the process
$$\int_{s}^t \frac{dW_u}{(t-u)^{\gamma}}$$ is not a Itô diffusion. Moreover, the singularity of the kernel function at $t$ does not help either, any approach I have tried seems to fail at some point.
However my intuition tells me that $$\int_{s}^t \frac{dW_u}{(t-u)^{\gamma}}\bigg|\mathcal{F}_s\sim N\left(0,\frac{(t-s)^{1-2\gamma}}{1-2\gamma}\right)$$ since the Brownian $W_t$ seems not to be affected by the filtration $\mathcal{F}_s$.
I would appreciate any help to determine if there exists an analitical solution
Many thanks!