Couldn't a public vote be forced to settle it once and for all, like in the Canadian House of Commons???
It may force the "undecided" to take sides and either elect a Prime Minister or elect a new Parliament all over again...
Ahh, the beauty of parliamentary democracy!!!
"The Public" has no say in the matter. They are simply tasked, occasionally, to voice a general party preference. After that, it is up to the party to decide.