Stronger linear lower-bound conjecture for the theta-function arithmetic sum

For positive integers $h$ and $k$, define

$$S(h,k):=\sum_{j=1}^{k-1}(-1)^{j+1+\lfloor hj/k\rfloor},$$

where $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$, and define

$$S(k):=\sum_{h=1}^{k-1}S(h,k).$$

Stronger linear lower-bound conjecture. For each prime $k>3119$,

$$S(k)>3k.$$

This is a further numerical strengthening of the preceding lower bound. The source reports that the primes up to $10000$ failing $S(k)<3k$ have largest value $3119$, but gives no proof of the conjecture.