The relationship between HDI and suicides is not very clear. However, lets try to look at the effect of increase in HDI on suicides in two strong economies- Japan and United States.
The graph for USA shows that initially, as HDI increased, the suicides increased till a point and then there was a sudden drop. Post that, it again started increasing with HDI.
The curve for Japan keeps rising and falling periodically. However, the recent trend has been a decreasing one.
Hence, we cannot say with certainty that suicides decrease with increase in HDI or vice versa. It varies with country.
One reason for increase in suicides with increase in HDI can be due to excessive work pressure and continuous effort to keep up with the upgraded lifestyle. However, this has not been strongly established yet.