The form is quite comprehensive. Completing it might resolve the issue.
Considering how many of these I have filled out, I would have the concern that filling it out would make the new head of the company even more nervous.
Personally, I stand by the previous advice that I gave on the subject. The OP has been using the company's equipment because it is neither cost effective nor practical for him (or any other potential contractor) to provide the same equipment themselves.
That logic overrides the reason why normally using the company's equipment would be indicative of employee status.