Let $(R, \mathfrak{m})$ be a Cohen-Macaulay ring, let $f_1, \dotsc, f_d \in \mathfrak{m}$ be a regular sequence, and let $n_1, \dotsc, n_d > 0$ be weights (feel free to assume that $n_1 = 1$ if it helps). I would like to consider the graded $R$-subalgebra $R' \subset R[T]$ generated by the $f_iT^{n_i}$ and form the "weighted blowup" $X = Proj(R')$. Is $X$ Cohen-Macaulay? If $n_1 = 1$ (the only case I care about), is at least the locus of $X$ where ``$f_1T$'' does not vanish, i.e., the last (first?) affine coordinate patch, Cohen-Macaulay?

I suspect that the answer may be contained in the commutative algebra literature and that maybe even $R'$ itself is Cohen-Macaulay--this would be even better--but I do not know where. For instance, in the case $n_1 = ... = n_d = 1$, when we are talking about the usual blowup / Rees algebra, the Cohen-Macaulayness of $R'$ was proved by Barshay in 1973.

I would be very grateful if someone could point me to the relevant literature, I know that Cohen-Macaulayness of various graded rings has been studied extensively, but I don't know where to look for this particular question. I tried using Hyry's paper on Cohen-Macaulayness of multigraded rings implying that of the diagonal subring, but its results did not seem to help directly (because of generation in degree $1$ assumptions).