Conjectured bounds relating quantum and private capacities

Let ${\cal N}$ be a quantum channel. The quantities $Q^{(1)}({\cal N})$ and $P^{(1)}({\cal N})$ denote the single-use quantum and private capacities, while $Q({\cal N})$ and $P({\cal N})$ denote their regularized capacities. Let ${\cal N}^c$ be a complementary channel, and let $P({\cal N}^c)$ be the regularized private capacity of the complementary channel.

Capacity bounds conjecture. For a quantum channel ${\cal N}$,

$$Q^{(1)}({\cal N}) \leq Q({\cal N}) \leq Q^{(1)}({\cal N}) + P({\cal N}^c),$$ $$P^{(1)}({\cal N}) \leq P({\cal N}) \leq P^{(1)}({\cal N}) + Q({\cal N}^c) + P({\cal N}^c) \leq P^{(1)}({\cal N}) + 2P({\cal N}^c).$$

These inequalities would sharpen the theorem preceding the conjecture by replacing the auxiliary quantity $M({\cal N}^c)$ with the private capacity $P({\cal N}^c)$; their validity would give more direct bounds on regularized capacities in terms of single-use capacities and the complementary channel.