Harshitha's asymptotic conjectures for alternating trigonometric sums

Let

$$S_1(k):=\sum_{j=0}^{2k-1}(-1)^j\sin^2\left(\dfrac{(2j+1)\pi}{8k+2}\right),\qquad S_2(k):=\sum_{j=0}^{2k}(-1)^j\sin^2\left(\dfrac{(2j+1)\pi}{8k+6}\right).$$

Harshitha's conjectures. The limits satisfy

$$\lim_{k\to\infty}S_1(k)=-\dfrac12,\qquad \lim_{k\to\infty}S_2(k)=\dfrac12.$$

These are asymptotic evaluations of alternating finite trigonometric sums arising from Ramanujan's theta functions; the supplied text attributes them to the authors of the cited work, but gives no resolution status.