Yes humble pie time.
You did say rectangular in the first post not square so I misread it from the start.
Never had a problem with any fuel pump other than not working due to a stuffed diaphragm and never had a problem with the round pump apart from the one clogged up with gum.
A leak in the head gasket will also make the engine difficult to start cold but start quite fine when hot as will a leaking valve.
To isolate the problem try using an alternative fuel tank suspended as high as you can over the engine to give the best pressure head possible.
If the problem persists then the problem is not with the supply pressure which as you have already been told is in the order of 2 PSI.
As for sarcasim, yes I tend to be direct, it is a cultural thing we don't go tippy toeing around down here smiling to some ones face while cursing them behind their backs.
I can and have been quite wrong and will be wrong in the future but 2 others have also mentioned it is unlikely to be your pump and we all know exactly how these impulse pumps work.
Sometimes it takes getting some ones back up to force them to take a step back from whatever is fixating their attention
As for you Husqvarna if it is fitted with an Intek then there is a better than average chance it has a leaky head gasket, I come across the odd one that had been leaking from new.
If it is fitted with a Kohler Courage then again good chance it is leaking pressure for the top plate which comes loose at the 4 bolts near the cylinder.
If it has a different model engine I have no common hard start problems other than the control cable slipping which has already been mentioned.
When I took over this workshop there was no spare fuel pumps so I bought 10 of them being sure they would fail regularly I still have 9 wasting space on the shelf.
There is around 400 mowers in the service run and around 350 of them run the current spec round fuel pump and the only one that needed replacing was simply gummed up.
Believe it or not most who are on this list are here to either be helped or to help others or both and knocking yourself out to fix a problem that you don't actually have is something that we collectively try to prevent.
Things like leaking head gaskets, if caught early are an easy & relatively cheap fix but if left unattended for extended periods can be very expensive.
Just finished doing a mini digger that had a loose head bolt originally but this escallated to a $ 1100 repair because nothing was done till the engine would no longer start USING STARTER FLUID .
The small leak in the head had erroded a deep groove in both the cylinder & head eventually burning out 3" of the head gasket. This caused the head to run hot requiring a complete replacement and the block has to be welded then machined and a new piston.
To the user, these thing are like old age, they slowly creep up on you, the engine gets harder & harder to start, but it always starts in the end & the engine gets louder & louder but unless you go directly from a new one to an old one you don't notice because it is happening slowly.