Re-flipping tails only in pairs with four coins

Four fair coins are tossed together. Afterwards, as long as at least two coins show tails, you may pick two tails-up coins and toss that pair again; a single coin can never be re-tossed by itself. You keep doing this until fewer than two tails remain. What is the expected number of heads at the end?

$88/27$. Condition on the first toss. With $3$ or $4$ heads nothing changes. With two tails you re-toss until the pair is not $TT$, ending at $3$ or $4$ heads with odds $2:1$, worth $\tfrac{10}{3}$. Working the same recursion back gives $\tfrac{29}{9}$ from one head and $\tfrac{88}{27}$ from none, and weighting by $\text{Binom}(4,\tfrac12)$ gives $\tfrac{88}{27}$.