Let

$R = \mbox{diag} (r_1,\dots,r_n)$, where $r_1, \dots, r_n > 0$, be a (positive) diagonal matrix.

$1_n \in \mathbb{R}^n $ denote the $n$-dimensional vector of all ones.

$S$ be a matrix defined by $[S]_{jk}=\sin (\theta_{j}-\theta_k +\delta)$, where $[\theta_1, \dots, \theta_n]^\mathrm{T} \in \mathbb{R}^n$ and $\delta \in \mathbb R$.

Is it true that $$\left( 1^\mathrm{T}_n S^T R S 1_n -1^\mathrm{T}_n R S R^{-1} S 1_n \right) \geq 0$$ always holds?

My thoughts so far:

Since $S^T R S = (R^{1/2} S)^\mathrm{T} (R^{1/2}S)$, therefore it is positive semidefinite and so $1^\mathrm{T}_n S^T R S 1_n \geq 0$.

Also, I see that when $1^\mathrm{T}_n S^T R S 1_n = 0$, then $1^\mathrm{T}_n R S R^{-1} S 1_n =0$.