The following infinite nested radical $$\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}$$ is known to converge to $\phi=\displaystyle\frac{\sqrt{5}+1}{2}$.

It is also known that the similar infinite nested radical $$\sqrt[3]{\frac{1}{2}+\sqrt[3]{\frac{1}{2}+\sqrt[3]{\frac{1}{2}+\sqrt[3]{\frac{1}{2}+\cdots}}}}$$ converges to $\displaystyle\frac{1}{t_3-1}=1.19148...$, where $t_3=1.83929...$ is the tribonacci constant, corresponding to the ratio to which adjacent tribonacci numbers tend. Note the similarities with the previous case, since $\displaystyle\frac{1}{\phi-1}=\phi$.

Interestingly, assuming that the degree of the radicals is $n$ and that the constant term under the radicals is $1/(n-1)$, the relationship between this type of nested radicals and the n-bonacci constants is valid even for the degenerate 1-bonacci numbers (whose sequence reduces to a series of 1). In this case, the radicals are canceled out, the constant term becomes $1/0=\infty$, and the result is $\infty$, which is in accordance with $\frac{1}{1-1}=\infty$.

Based on these considerations, it could be hypothesized that similar nested radicals may exist for higher order n-bonacci numbers. For example, for $n=4$, we could expect that a nested radical giving $\displaystyle\frac{1}{t_4-1}=1.07809...$, where $t_4=1.92756...$ is the so-called tetranacci constant (limit of the ratio between adjacent 4-bonacci numbers), might be

$$\sqrt[4]{\frac{1}{3}+\sqrt[4]{\frac{1}{3}+\sqrt[4]{\frac{1}{3}+\sqrt[4]{\frac{1}{3}+\cdots}}}}$$

Unfortunately, in this case the radical seems not to work (it gives $1.09279...$ instead of the expected $1.07809...$). Also, I was not able to find any rational number that, inserted in this nested radical as the constant term, satisfies the equivalence (the numerical value that works is $k=0.27282...$). Similar results are obtained for higher values of $n$. Which is the reason for this? Is there any way to prove or disprove the existence of similar nested radicals linked to n-bonacci numbers for $n>3$?