The polls suggest that Joe Lieberman is capable of winning the Democrat primary in Connecticut, and then being re-elected rather comfortably in November. Certainly, polls indicated a competitive primary for him, and it is probably getting closer, but sitting Senators are not often dumped by their own parties, especially not former Veep candidates. And yet . . .
If Joe Lieberman was absolutely unwilling to run for Senate as an independent or a Republican, why will he not say so? It could be that he is trying to threaten primary voters with the prospect that he will run - and very possibly win - against Ned Lamont if Lamont is chosen as Democrat candidate. But if so, it is a badly thought out strategy. It is likely to infuriate Democrat primary voters and make them more inclined to vote for Lamont. It is reasonable to assume that Lieberman is leaving the option open because he is genuinely considering it.
But this is where it gets awkward. As blogger DavidNYC points out in The Swing State Project the calendar simply does not allow Lieberman to keep both options - Democrat or independent - open. He would need to gather enough signatures to qualify for the ballot in a single day, if he waited until after the result of the Democrat primary.
If he is serious about the option of running as an independent he will have to start gathering signatures well before the August primary. He may even have started already, though it is difficult to believe he will get very far with this without the secret blowing.
My own guess: he will watch the polls. If he is in real doubt about winning the Democrat primary he will jump ship and start an independent bid some time in June or July.