I've looked at some highly composite numbers and realized that a lot of them are almost primes, i.e. differ only by $1$ to the next closest prime. Here's a short list (I made) of highly composite numbers and whether they are almost prime: $$ 1 \text{, true $(+1)$} \\ 2 \text{, true $(+1)$} \\ 4 \text{, true $(\pm 1)$} \\ 6 \text{, true $(\pm 1)$} \\ 12 \text{, true $(\pm 1)$} \\ 24 \text{, true $(-1)$} \\ 36 \text{, true $(+1)$} \\ 48 \text{, true $(-1)$} \\ 60 \text{, true $(\pm 1)$} \\ 120 \text{, false} \\ 180 \text{, true $(\pm 1)$} \\ 240 \text{, true $(\pm 1)$} \\ \dots $$
I've thought about it, and it's actually not so surprising that many highly composite numbers (which I've checked) are almost prime. Since $n \pm 1$ won't share any prime divisors with $n$, it's likely for $n \pm 1$ to be prime if $n$ has a high number of divisors.
But are there infinitely many highly composite numbers that are $\pm 1$ a prime? I don't know how I would prove something like this—I'm not even sure if this conjecture really holds.
Also, if there are infinitely many such numbers, do they get rarer? And at what rate?