I'm inclined to say "yes" to $T_1>T_2$ and no to $P_1<P_2$.
For simplicity, suppose the monopolist's marginal cost is constant. Then consumer surplus is the area under the (inverse) demand curve and above the marginal cost. In a two-part tariff scheme, the monopolist would charge entrance fee equal to consumer surplus and unit price equal to marginal cost.
If, in your language, consumers with "high demand" are understood as those who are willing to pay more for each unit of the good than those consumers with "low demand", then high demand consumers generate higher consumer surplus than the low demand ones.
Suppose monopolist can distinguish between the two types and offer different two-part tariff bundles to each (e.g. a theme park company operating in US and India charging different prices in the two countries). Then it will be the case that the monopolist charges a higher entrance fee to the high type (because they generate higher consumer surplus) but the same unit price for both types (because this is related to the marginal cost of production, which is the same regardless of the consumer types).