I'd like some advice regarding the following question, which I have been struggling with for long time.

Let's call the shaded region in the below $S_3$. It is the union of three congruent isosceles triangles with height $1$ and bottom side length $1/3$. Likewise, we call $S_n$ the region of $n$ congruent isosceles triangle with height $1$ and bottom side length $1/n$. Let $\{n_i\}$ be an arbitrary finite subsequence of $\mathbb{N}$ with $k$ elements. I'd like to prove that $$h(\cap_{i=1}^k S_{n_i})\geq \frac{2}{k+1}, \forall k\in\mathbb{N}$$ for an arbitrary $\{n_i\}$, where $h$ maps the set to the height of the set. For example, in the above case $S_2\cap S_4$, the height is $2/3$, so the above inequality obviously holds.

In fact, it was proved that the above inequality holds for $k$ from 1 to 7, and I showed numerically that the inequality holds for numerous possible sequences $\{n_i\}$ for some higher $k$. It was also proved that, if the right side of the inequality is replaced with $\frac{2}{2 k-1+1/(2k-3)}$, the inequality holds for an arbitrary positive integer $k\geq 4$. Also, One can assume the following without loss of generality: $$\gcd(n_1,n_2,...n_k)=1\\(k+1)|n_1$$

**Added:**

- It was proved that, if $\{n_i\}$ is an arithmetic progression, the inequality holds for any positive integer $k$.
- Clearly, at least one of $n_i$ is even for non-trivial cases, and therefore the inequality holds for the sequence of any prime numbers.