2025-04-14 Update Part 2: I have been able to improve the constant to $3 \log_2 n / \log_2 (27/23) \approx 12.97$ from the previous $~15.57$.

2025-04-14 Update: There was an error in my previous writeup. It has been fixed in the version below. Kyle also shared with me that this problem is an example of non-adaptive group testing, a well-studied subject. I also improved the constant from $4\log_2 n / \log_2(8/7) \approx 20.76$ to $3\log_2 n / \log_2(8/7) \approx 15.57$, by improving the second union bound

The past few days I spent at the 44th Midwest Probability Colloquium, and there I met another graduate student interested in good latex setups. From our conversation on the subject I figured this may be a good opportunity to put my current setup on my website.