I'm not a practicing lawyer but I went to law school. I have no legal experience in your state, this is just a hypothetical, and this is not legal advice, yadda yadda yadda.
I'd say it is extremely unlikely. A business venture that requires a substantial cash infusion that you refuse to put in would not "pierce the veil" or otherwise there would be no sense to having a "veil" in the first place. If you follow the correct corporate formalities (this is key!) then piercing the veil should only be for "bad boy" acts which are "malfeasance" or bad-intentioned. Saving money (or deciding when to spend and when to "throw in the towel," so to speak) should be a "business judgment."
The most common reason for "piercing the veil" is not following the proper corporate formalities and treating the company like your personal ... thing. If you treat it like there is no "veil," the court won't step in to provide one.
On the other hand, as an example, if you withdraw money and leave the company too thinly capitalized and it fails, it is still unlikely to pierce the veil unless you really did something "bad" and the court thinks it should "claw back" some of your ill-gotten gains. This is not close to the situation you hypothesized.
IF you want a better answer, you'll have to pay a lawyer to look up actual real case law in your state and even then the answer will still seem grey. Corporate law is state-specific and works by precedent so that means research is needed and you will always find some case that leaves you scratching your head thinking, "what the heck were they thinking?" and the answer is, "sometimes cases are just wrongly decided." Going to court is always a gamble.