Sequence of product of sums and reciprocal sums

Similar to the last two problems, but no triangles this time.

Suppose $t_1, t_2, \dots, t_n, \dots$ are a sequence of positive reals.

Let

$$S_n = \sum_{i=1}^{n} t_i = t_1 + t_2 + \dots + t_n$$

and

$$R_n = \sum_{i=1}^{n} \frac{1}{t_i} = \frac{1}{t_1} + \frac{1}{t_2} + \dots + \frac{1}{t_n}$$


Show that:

If $$S_k R_k \lt k^2 + 1$$ for infinitely many $k$, then all the $t_i$s must be equal.