Conjecture on representations by three signed triangular-number forms

Let $\mathbb{N}_0$ denote the set of nonnegative integers. Consider the two forms

$$a(2a-1)+b(2b-1)+c(2c+1)$$

and

$$a(2a-1)+b(2b+1)+c(2c+1),$$

where $a,b,c\in\mathbb{N}_0$. Three-form representation conjecture. Every natural number other than $8$ and $68$ can be written in at least one of these two forms. This conjecture proposes a near-universal representation theorem for natural numbers by combinations of triangular-number expressions; the stated exceptions are the only excluded values.