Going-up spectrum realization conjecture for simultaneous Diophantine exponents

Let $(\lambda_N)_{N\geq 1}$ and $(\widehat{\lambda}_N)_{N\geq 1}$ be non-increasing sequences of real numbers. Assume that, for every $N\geq 1$, $\widehat{\lambda}_N\geq 1/N$ and

$$\widehat{\lambda}_{N}+\frac{\widehat{\lambda}_{N}^{2}}{\lambda_{N}}+\cdots+\frac{\widehat{\lambda}_{N}^{N}}{\lambda_{N}^{N-1}}\leq 1.$$

Assume also that, for all $k\geq n\geq 1$,

$$\lambda_{k}\geq\frac{n\lambda_{n}+n-1}{n(k-1)(\lambda_{n}+1)+1},\qquad \widehat{\lambda}_{k}\geq\frac{n+\widehat{\lambda}_{n}-2}{(n-1)(k-1)}.$$

Going-up spectrum realization conjecture. There is $\underline{\xi}\in\mathbb{R}^{\mathbb{N}}$ such that, for every $N\geq 1$, its first $N$ coordinates $\underline{\xi}_N$ satisfy

$$\lambda_N(\underline{\xi}_N)=\lambda_N\quad\text{and}\quad\widehat{\lambda}_N(\underline{\xi}_N)=\widehat{\lambda}_N.$$

This conjecture asks whether the displayed necessary restrictions on the sequences are also sufficient for simultaneous realization by the projections of one infinite real vector. It is motivated by the preceding going-up inequalities and is presented as an expected sufficiency statement; no resolution is supplied here.