**Counterexample.** Identify the points of the [Fano plane](https://en.wikipedia.org/wiki/Fano_plane) with the numbers $1,2,3,4,5,6,7$ and call the lines $\ell_1,\ell_2,\ell_3,\ell_4,\ell_5,\ell_6,\ell_7.$

Let $N=\{0,1,2,3,4,5,6,7\}.$ Let $P$ contain the $5$-element sets $x_i=N\setminus\ell_i$ and the $4$-element sets $y_i=\ell_i\cup\{0\}$ where $i=1,\dots,7.$ Conditions (1)–(4) are clearly satisfied.

Let $\alpha=\frac17\sum_{i=1}^71_{x_i}$ and $\beta=\frac17\sum_{i=1}^71_{y_i}.$

Then $\alpha_0=\beta_0=1$ while, for $1\le i\le7,$ we have $\alpha_i=\frac47$ and $\beta_i=\frac37.$