Venture Deals - Convertible Debt

Currently reading Venture Deals, and a statement the authors make in relation to convertible debt doesn't make any sense to me.

"By definition, if you raise a convertible debt round, your company is insolvent."

How does this compute? I realize that interest payments will draw down equity over the maturity of the debt by lowering net income and in turn retained earnings, thus potentially lowering equity to a negative value (ie. insolvency). But this statement seems to suggest insolvency as soon as the debt round is issued - how can this be when the cash and debt going onto the balance sheet are of the same value? Would it not have to be the case that a company was already insolvent before the debt issuance to be insolvent after it? And this doesn't even make sense because nobody is financing an early-stage venture in such a financial situation - besides the founders themselves via additional equity.

I do understand that when warrants are used to discount conversion of the debt to equity in the event of a later financing, the issue of OID - original issue discount - can arise. In that situation, the cash going on to the BS would be less than the debt (hence the discount), drawing down equity immediately and pushing a company towards insolvency. But in any case, this problem is apparently avoided by the lawyers/accountants making sure to attach a value to the warrants separate to the debt in the deal.

So ultimately, how can the authors make a blanket statement regarding insolvency here? Instinctively, it doesn't sound right, and further analysis doesn't clear up my confusion.

Any insights much appreciated.