This question concerns distributions $\mu$ over the naturals $\mathbb{N}=\{1,2,\ldots\}$. For $q\ge1$, let us define the $q$th moment of entropy: $$ H_q(\mu)=\sum_{i=1}^\infty \mu(i)|\log\mu(i)|^q, $$ so $H_1(\mu)$ is just the usual entropy.

I am interested in a sequence of distributions $\mu_n$ satisfying the following properties:

1. $\mu_n(1)\to1$ as $n\to\infty$
2. $\limsup_{n\to\infty}H_2(\mu_n)<\infty$
3. $\liminf_{n\to\infty}H_1(\mu_n)>0$

Does such a sequence exist? I'd be satisfied with mere existence (though a construction would, of course, be nice.)

If such a sequence does not exist (proof?), I'd be happy with the following weakening:

1'. $\mu_n(1)\to1$ at some fixed rate

2'. $H_1(\mu_n)\to0$ at an arbitrarily slow rate