Conjecture on singular regularisation parameters for multiple t-values

Let $(\lambda_i)_{i=1}^{\infty}$ be the sequence of singular regularisation parameters, beginning

$$(\lambda_i)_{i=1}^{\infty}=\left(0,2,\frac{28}{11},\frac{242}{91},\frac{64472}{23479},\ldots\right).$$

Conjecture on singular regularisation parameters. The sequence is increasing and bounded by $3$:

$$\lambda_{i+1}>\lambda_i,\qquad \lambda_i<3\quad\text{for all }i,$$

and it converges to $3$:

$$\lim_{i\to\infty}\lambda_i=3.$$

These parameters arise from singular regularisations in the motivic multiple $t$-value computations; the source reports numerical and structural evidence but no proof of these three properties.