This study focuses on people who have pulmonary hypertension with heart failure with preserved ejection fraction (PH-HFpEF), also known as diastolic heart failure or diastolic dysfunction. This is a type of heart failure in which the heart is stiffer and does not relax and fill appropriately, which can lead to increased blood pressure in the lungs. Patients who have this condition may feel short of breath or have difficulty doing physical activity. The purpose of the study is to evaluate an experimental drug called levosimendan in patients who have PH-HFpEF. The study has 2 stages. In the first stage, researchers will assess whether levosimendan, compared to placebo (inactive substance), improves the distance participants with PH-HFpEF can walk in 6 minutes. Participants will be randomly assigned to receive either levosimendan or placebo. In the second stage, all participants will receive levosimendan, and its effects and safety will be evaluated for a longer period of time. Stage 2 is optional. Levosimendan is approved for use as an infusion (intravenous injection) to treat hospitalized patients with breathing difficulties associated with a type of serious heart failure; however, it is considered experimental in this study because it has not been approved by the U.S. Food and Drug Administration (FDA) to treat PH-HFpEF or approved to be taken by mouth.